have value around J21bg ∼ 40 (Dijkstra et al. 2008), which is smaller than the critical intensities, especially in the case of a T5 background. However, the background will inevitably fluctuate spatially, and thus a fraction of all halos will be irradiated by an intensity exceeding J21crit. Then, the lower this critical intensity, the larger the fraction of halos which are suitable candidates for direct SMBH formation; e.g. according to the distribution proposed by Dijkstra et al. (2008), a decrease in J21crit from 104 to 103 means an increase in the fraction of irradiated Tvir≈ 104K halos from negligibly small (. 10−8) to ∼10−6 (see their Figure 2). For J21crit∼102, the halo fraction even increases to ∼10−3. With a sufficiently low J21crit, one could argue that this mechanism provided many, if not all, seeds for the SMBHs observed in galaxies today.
5.4 Fragmentation
In order to estimate the mass of the black hole seed, a criterion for when fragmentation occurs is required. It has been shown that the equation of state helps to determine how strongly self-gravitating gas fragments (e.g. Spaans & Silk 2000; Li et al. 2003).
A fragmentation criterion can be formulated in terms of the polytropic exponent (or effective adiabatic index), which can be expressed as:
γ = d ln P
d ln ρb (5.1)
= 1 + d ln T
d ln ρb (5.2)
Roughly speaking, fragmentation occurs efficiently when γ. 1, i.e. during temperature drops, and (almost) stops when the equation of state stiffens, so when γ& 1. Thus, the preferred mass scale of the fragments is set by the relevant Jeans mass at the density where the polytropic exponent increases above a threshold value γfrag ≈ 1 after a tem-perature minimum. Of course, this only holds provided that the halo is massive enough to be able to collapse. In theory γfrag= 1 and fragmentation stops approximately when the gas becomes isothermal, but based on the simulations of Bromm & Loeb (2003), Omukai et al. (2008) argue for a threshold value slightly below unity, so that fragmen-tation does not occur during the atomic cooling phase, where γ has been found to be between 0.95 and 1. For this reason a fragmentation threshold of γfrag= 0.95 is adopted here.
5.4.1 Without turbulence
The evolution of the polytropic exponent in a halo without turbulence or magnetic fields (or with an initial magnetic field strength . 0.01 nG) is quite simple, as can be seen in Figure 5.1. Initially, the temperature increases adiabatically and γ ≈ 5/3, until enough molecular hydrogen is formed and the gas starts to cool. The temperature minimum occurs at a number density of ∼1.0 × 103cm−3, which sets a fragment mass of
∼7.9 × 103M . At the end of the simulation the polytropic exponent is decreasing again, but after comparison with the results of similar simulations that go up to higher densities (e.g. Omukai et al. 2008; Schleicher et al. 2009) it does not seem likely that another
0.5 1
1.5 Without turbulence; B
0 = 0
0.5 1
1.5 Without turbulence; B
0 = 3 nG
0.5 1
1.5 Without turbulence; B
0 = 13 nG
10−1 101 103 105 107
0.5 1
1.5 With turbulence; B0 = 0
nb [cm−3]
Polytropic exponent γ
Figure 5.1 – Evolution of the polytropic exponent γ with baryonic density for several different models. The top three plots show the results for the zero-field, 3 nG, and 13 nG case of the model without turbulence, respectively, while the bottom plot shows the results for the zero-field case of the model with turbulence. The dashed horizontal lines indicate γ = 0.95 and 1; the dotted vertical line indicates when virialization occurs.
February 2013 5.4. FRAGMENTATION
fragmentation episode will occur, because eventually the fragments become completely opaque to their own cooling radiation and the collapse becomes approximately adiabatic.
For an initial field of 0.1 nG, the evolution of the polytropic exponent is rather similar to the zero-field case, but the temperature minimum is shifted to a lower density (∼4.2 × 102cm−3). As a result, the fragment mass is somewhat larger compared to the zero-field case, Mfrag≈ 1.2 × 104M . A similar shift of the temperature minimum to an even lower density (∼2.5 × 102cm−3) occurs also for B0 = 0.5 nG. Here, the minimum temperature reached by the gas is increased compared to the zero-field case, which would result in a somewhat larger Jeans mass. However, at this point in the evolution, the magnetic Jeans mass dominates over the thermal Jeans mass, and thus the fragment mass is increased greatly, Mfrag≈ 1.9 × 106M . Near the end of the simulation, the polytropic exponent stays approximately constant at ∼1.10. For B0= 1 nG, the temperature drops more steeply at first, but the location of the minimum does not change much compared to the 0.5 nG case. However, since the magnetic field is stronger, the magnetic Jeans mass and thus the fragment mass will be larger, Mfrag ≈ 5.2 × 107M . For B0 = 3 − 10 nG, the temperature also drops steeply at first, but the location of the minimum is close to that of the zero-field case. At this point, the magnetic Jeans mass dominates, and the fragment mass increases with increasing magnetic field; ∼3.5 × 109M , ∼2.1 × 1010M , and ∼2.5 × 1011M for B0 = 3 nG, 5 nG, and 10 nG, respectively. However, at the moment where the temperature instability occurs, the polytropic exponent shoots steeply up and down, thereby crossing the threshold value again. It is uncertain whether this will result in another fragmentation episode or not. However, it does not seem likely that fragmentation will occur, given that the time where γ < γfrag is shorter than the free-fall time at that point, and thus the change happens too quickly for the system to be able to react. Afterwards, the gas becomes nearly isothermal with γ ≈ 0.99. Finally, even for halos with B0 & 13 nG a (relative) temperature minimum occurs, albeit very shallow, at a density of ∼1.0 × 10−1cm−3, which corresponds to a fragment mass of
∼2.2 × 1011M . Afterwards, the gas becomes nearly isothermal with γ as above.
5.4.2 With turbulence
When turbulence is taken into account, the differences between different initial field strengths are much smaller. In the zero-field case, the thermal evolution is quite dif-ferent from the zero-field-zero-turbulence case, but the temperature minimum occurs at a similar density, ∼7.2 × 102cm−3. The minimum temperature reached by the gas is however larger, which results in a larger fragment mass, ∼3.4 × 104M . Afterwards, γ flattens off and stays approximately constant at ∼1.08. When a magnetic field is included, the location and value of the temperature minimum are very similar to the zero-field case, and at this point the thermal Jeans mass always dominates over the magnetic Jeans mass. Thus, the fragmentation behavior is nearly unchanged.
5.4.3 General fragmentation behavior
The fragmentation behavior is affected by the following factors: the minimum tempera-ture, the location of the temperature minimum, the absence of a temperature minimum when the gas is cooled by atomic hydrogen and evolves (nearly) isothermally, and the
magnetic Jeans mass (and thus the magnetic field strength) in the case where it domi-nates over the thermal Jeans mass. Of course, fragmentation does not occur when the fragment mass is larger than the mass of the halo.
When only one of these factors varies, it is easy to predict the outcome. For example, the minimum temperature increases with increasing injected velocity while the location of the minimum stays unchanged, as can be seen from Figure 4.25, as long as vinis larger than 3% of the virial velocity. Since the magnetic Jeans mass is not important here, an increase in injected velocity thus means an increase in fragment mass.
However, when two or more of these factors change simultaneously, the result is not so straightforward. Both the minimum temperature and the density at which this minimum occurs are, for example, altered by a radiation background. The stronger the UV intensity, the higher the minimum temperature, which gives rise to higher fragment masses, but also the higher the density at which this minimum occurs, which gives rise to lower fragment masses if the thermal Jeans mass dominates. This is the case if turbulence is important or if the magnetic field is small. Because these two factors counteract, it becomes necessary to compute the fragment mass for each case. It turns out that the effect of the density is the strongest, and the fragment mass decreases with increasing UV intensity (but stays within the range ∼104M ∼ 105M ), as long as molecular hydrogen cooling still becomes important. However, if the magnetic Jeans mass dominates, which occurs in halos where turbulence is not important and B0 &
0.5 nG, the fragment mass actually does increase with increasing UV intensity, due to the effects of delayed molecular hydrogen formation on the magnetic field strength. For example, for B0 = 1 nG, the fragment mass increases from ∼5 × 107M without a radiation background, to ∼2 × 108M for a T5 background with J21= 104.
Atomic cooling halos, in which the gas evolves nearly isothermally (γ ≈ 0.99) once it has heated up to ∼104K, do not appear to go through a temperature minimum, see e.g. Figures 4.16 and 4.17. However, after the gas heats up, it cools down again by a small amount afterwards. This results in a dip in the polytropic exponent and thus the gas could become susceptible to fragmentation. Turbulent halos that cool through atomic hydrogen as a result of their large mass, or because they are exposed to a radiation background with an intensity larger than J21crit, could fragment around a density of ∼10−1cm−3, which results in a fragment mass of ∼108M . For halos without significant turbulence in the presence of a supercritical radiation background, the density at which fragmentation could occur depends on the magnetic field strength, as halos with a stronger field heat up faster. In the zero-field case, the gas could fragment around a density of ∼102cm−3, which results in a fragment mass of ∼106M . For larger fields, fragmentation shifts to smaller densities, with a minimum density of ∼10−1cm−3 corresponding to a fragment mass of ∼108M if the thermal Jeans mass dominates, and larger but dependent on the magnetic field strength if the magnetic Jeans mass dominates.
5.4.4 When is the fragment mass largest?
The fragment mass is increased to ∼106M or more (as opposed to the ‘standard’
∼104M ) for the parameter ranges described hereafter. A parameter range is denoted as “(T or NT, B0, J21subcrit or J21supercrit)”, where T stands for halos with turbulence